Is it possible to use the Extrude or Spin tool to get something to rotationally extrude along a curved path?
I am not sure I understand what you mean, is this the shape you wish to achieve?



The way I see it you can do it in several distinct ways


Either extrude your whole shape at once and bevel the outer edges of the result afterwards. You can use a mirror modifier in two axis so you only have to model half of it.
Draw the quarter circle shape section as a mesh and extrude it up step by step along those selected edges. You may use snapping to guide your shape all the way up, or
Draw both your section (semi circle) and the path (curved shape) of the bow as a bezier curve, use the section as a bevel object on the bow shape bezier curve. Use the curve radius on a per vertex basis to control the thickness of the bow along the path. (Option I used in the image above and bellow)